Note that the hypotheses of this theorem guarantee that \(N\)  acts freely on the specified tree and in particular \(N\)  is necessarily a free group. Note also that, according to [1]}, if \( b^{(2)}_1(G)>0\)  then \(G\)  has no commensurated infinite amenable subgroup and according to [2]} does not have property (T). If we also have \(b_2^{(2)}(G)=0\) , then \(G\)  is in the class \(\mathcal {D}_{\rm reg}\)  by [3]}. We refer the reader to [4]} for background on property (T) and to [3]} for the definition of the class \(\mathcal {D}_{\mathrm {reg}}\) .
By the main result of Osin's paper [6]} we have the following corollary.
